Bayesian Detection

The Bayesian approach to the problem of detecting changes in distributions, can be described in the following terms. Suppose that we decide to monitor the stability of a process with a statistic $T$, having a distribution with p.d.f. $f_T(t;\theta)$, where $\theta$ designates the parameters on which the distribution depends (process mean, variance, etc.). The statistic $T$ could be the mean, $\bar X$, of a random sample of size $n$; the sample standard-deviation, $S$, or the proportion defectives in the sample. A sample of size $n$ is drawn from the process at predetermined epochs. Let $T_i$ $(i = 1,2,\cdots)$ denote the monitoring statistic at the $i$-th epoch. Suppose that $m$ such samples were drawn and that the statistics $T_1,T_2,\cdots,T_m$ are independent. Let $\tau = 0,1,2,\cdots$ denote the location of the point of change in the process parameter $\theta_0$, to $\theta_1 = \theta_0 + \Delta$. $\tau$ is called the change-point of $\theta_0$. The event $\tau = 0$ signifies that all the $n$ samples have been drawn after the change-point. The event $\tau = i$, for $i = 1,\cdots,m-1$, signifies that the change-point occurred between the $i$-th and $(i+1)$-st sampling epoch. Finally, the event ${\tau = m^+}$ signifies that the change-point has not occurred before the first $m$ sampling epochs.